We prove robustness results in three settings: (i) the Dirac threshold for containing a perfect matching in a \(k\) -uniform hypergraph, (ii) the Hajnal–Szemerédi minimum degree condition for containing a \(K_r\) -factor, and (iii) the minimum degree condition for containing bounded-degree spanning trees. It is worth mentioning that finding the probabilistic thresholds for these properties in random subgraphs of the complete (hyper)graph were major open problems in the field. The thresholds for (i) and (ii) were found by Johansson, Kahn, and Vu [1]} and the threshold for (iii) was found by Montgomery [2]}. The fact that we are able to consider the minimum-degree setting for all three in just one paper speaks to the strength of the spread-measure results in [3]}.
